Very wise to have kept the Extended Warranty John. As I did too.
Can you Photograph or Video the Black screen next time the problem occurs and show them the image?
Ask them to keep the car for a couple of days, so that they can experience the fault as it occurs ?
It cannot be Rocket Science to isolate the problem between Camera and Reversing switch ?
Any Lexus Indies locally ?
1. Toyotatech, http://www.toyotatech.co.uk/
Coltsford Drive Guildford, GU1 1YG. Surrey Tel;07833 456051
2.
2. Westfield Motors, www.westfieldmotors.co.uk/
Rawreth Ind Est, Rayleigh, Essex. Tel; 01268 783205